On stabilizers in finite permutation groups

Abstract

Let $G$ be a permutation group on the finite set $\Omega$. We prove various results about partitions of $\Omega$ whose stabilizers have good properties. In particular, in every solvable permutation group there is a set-stabilizer whose orbits have length at most $6$, which is best possible and answers two questions of Babai. Every solvable maximal subgroup of any almost simple group has derived length at most $10$, which is best possible. In every primitive group with solvable stabilizer, there are two points whose stabilizer has derived length bounded by an absolute constant.